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Civil
Prop reaction from an intermediate point load
Upward reaction at the prop B of a propped cantilever (clamp at A) when a point load P acts at distance a from the clamp, 0 ≤ a ≤ L.
- Ref
- CIV-24
- Rev
- v1
- Field
- Civil
- Result
- RB [N]
For a point load P at distance a from the clamp of a propped cantilever, compatibility at the prop gives RB = Pa²(3L − a)/(2L³). The prop takes none of the load when a = 0 and all of it when a = L. Between these the fraction a²(3L − a)/(2L³) is a smooth curve that rises steeply toward the prop.
Upward reaction at the prop B of a propped cantilever (clamp at A) when a point load P acts at distance a from the clamp, 0 ≤ a ≤ L. At a = L the prop takes the whole load. Add other contributions by superposition only while every support stays engaged.
Calculator
- Fraction of the load taken by the prop0.3125
- Load carried by the clamp, P − RB13,750 N
- Clamp moment, Pa − RB·L15,000 N·m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The prop takes 31.25 % of the 20 kN load, 6.25 kN, with the clamp taking 13.75 kN.
- At midspan (a = L/2) the prop would take 31.25 % of the load, so even a mid-length load is carried mostly by the clamp.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| P Point load | 100 to 100,000 N | 31.25 | 31,250 | 31,220 | +10% | +10 % |
| a Load distance from the clamp | 0.1 to 4 m | 18.59 | 20,000 | 19,980 | +10% | +18.6 % |
| L Length | 2 to 20 m | 290 | 20,000 | 19,710 | +10% | −15.9 % |
Sweep of Load distance from the clamp
| Load distance from the clamp (m) | Prop reaction (N) |
|---|---|
| 0.1 | 18.59 |
| 0.5875 | 615.5 |
| 1.075 | 1,973 |
| 1.563 | 3,982 |
| 2.05 | 6,534 |
| 2.538 | 9,520 |
| 3.025 | 12,830 |
| 3.513 | 16,360 |
| 4 | 20,000 |
Sweep one variable
Variables and how each one contributes
Prop reaction
What the equation gives you. Everything else feeds it.
Point load
The point load, in N. Scales the prop reaction in direct proportion. Raising it by 10 % (to 22,000) moves the result by +10 %, from 6,250 to 6,875 N.
Load distance from the clamp
The distance of the load from the clamp, in m. Moving the load toward the prop raises the share it takes, with a nearly quadratic rise. Raising it by 10 % (to 2.2) moves the result by +18.6 %, from 6,250 to 7,411 N.
Length
The beam length, in m. A longer beam means the same distance a is a smaller fraction of L, lowering the share. Raising it by 10 % (to 4.4) moves the result by −15.9 %, from 6,250 to 5,259 N.
Worked example with the values above
- Fraction at the prop: a²(3L − a)/(2L³) = 4 × 10 / (2 × 64) = 0.3125.
- Prop reaction: P × fraction = 20,000 × 0.3125.
- RB = 6,250 N.
Assumptions and limits
- Rigid supports with no settlement, and a prismatic beam of constant EI.
- Linear elastic behaviour, so compatibility and superposition apply.
Common mistakes
- Treating a flexible support as rigid.
- Applying a formula for uniform load to a different load pattern.
- Forgetting that the sign convention makes a support moment negative when sagging is positive.
Statistics and uncertainty
Redundant reactions depend on the stiffness of the supports being as assumed. A prop that settles by only a few millimetres, or a clamp that is not fully rigid, can change the reaction by 10 % or more, so these ideal results are best read as limits on the real values.
Where it is used
- Sizing props, continuous beams and fixed-ended members.
- Setting up moment distribution analysis of frames.
Maths you need
- Compatibility: deformation at a redundant support must match the real support condition
- Fixed-end moment: the moment at a clamped end under load
- Rotational stiffness: moment needed to rotate a joint by one radian